As a mathematician, something that I always envied physicists is the uninhibited way they use mathematics.

The classic example is the Dirac delta function, which is a function that’s zero everywhere except the origin, but has area one. The fact that no such function exists is only a minor inconvenience. Delta functions can be made rigorous as a distribution, but the concept well predates its formal definition. For example, Green’s functions, which are defined in terms of the delta function, date from 1828.

A more dramatic example is the concept of a dipole. A dipole is the limit of two electric charges of opposite charge as the distance between them goes to zero. It can also be thought of as the limit of the difference of two Dirac delta functions, or even the derivative of a delta function. Dipoles are used to approximate the effect of magnets from a long distance. In terms of distributions, a dipole is the derivative operator on the space of smooth functions, but this is far from the physical intuition.

Try some String Theory seminars. When I was studying mathematics years ago I used to hang out with theoretical physicists. I couldn’t believe the amount of high powered mathematics they’d throw around: algebraic geometry, esoteric (co)homology theories, modular forms, representation theory, category theory and so on. They’d use something like Grothendieck-Riemann-Roch as if they’d been using it since birth and yet they barely knew the foundations of algebraic geometry. I could never figure it out. In fact, I was never sure whether they really understood what they were talking about at all

Do you know how they think of Grothendieck-Riemann-Roch? String theorists must have some sort of intuitive picture.

The subject matter of GRR is the bread and butter of physics – vector bundles over manifolds. But I can’t really say more than that. Presumably there

isa way to visualise it that doesn’t require you to read several hundred pages of densely written material on on commutative algebra and schemes, but people don’t talk much about how they visualise things